7  untyped-comb and DAG Reduction

7.1 The Combinatory Architecture

The untyped-comb module handles the transformation of verified logic into an executable format. The system structurally translates the Abstract Syntax Tree (AST) away from dependent types into a flat, variable-free combinatory representation.

#| label: comb-inductive-type
#| eval: false

inductive Comb
  | K : Comb 
  | S : Comb 
  | I : Comb 
  | U : Comb 
  | var : String -> Comb
  | app : Comb -> Comb -> Comb
  | t_inject : Comb -> Comb
  | lazy_thunk : Comb -> Comb
  | terminal : String -> Comb

This compilation model replaces hierarchical memory structures with precise routing combinators. Variables and substitution environments are removed from the operational layer, replaced entirely by the structural fusion and duplication rules dictated by the \(S\) and \(K\) execution nodes.

7.2 Directed Acyclic Translation

To safely process highly entangled combinator streams on bare-metal hardware, the execution engine translates the operations into a Directed Acyclic Graph (DAG).

The compiler executes an \(O(V+E)\) topological sort utilizing Kosaraju’s algorithm to analyze the semantic graph. By identifying 0-weight semantic cycles (Strongly Connected Components), the engine safely projects non-well-founded referential loops without triggering memory exhaustion.

7.3 Stabilizing the Execution Layer

Because the nf-sketches architecture evaluates an unstratified universe natively, it requires specific mechanical safeguards to manage execution limits structurally:

  • Algorithmic T-Weaking: The compiler utilizes the integer distances generated by the static verification matrix to dynamically inject \(T\)-operators (Comb.t_inject). These topological markers stabilize the geometric mapping of operations, acting as self-regulating type-shifts natively within the unstratified graph.
  • MCM Bounds: The Minimum Cycle Mean is utilized to calculate dynamic operational limits across semantic boundaries, freezing unresolvable calculations safely as terminal structures.
  • Backend Targeting: Once formatted as an acyclic combinatory DAG, the data is optimally structured for parallel, high-performance evaluation inside topological frameworks like Interaction Nets or HVM2, bypassing standard von Neumann memory management overhead.